A Fourier spectral method for the nonlinear coupled space fractional Klein‐Gordon‐Schrödinger equations

Abstract

In this paper, we give an efficient numerical method for the nonlinear coupled space fractional Klein-Gordon-Schrödinger (NCSFKGS) equations, based on the Crank-Nicolson method, the central difference method and the Fourier spectral method. As far as we know, no one has studied the Equations (5)-(6) in our paper, these equations are different from those in [39] which only considers space fractional Schrödinger equation while the Klein-Gordon equation is classical, here, we consider the two equations which are both space fractional. In this paper, the Crank-Nicolson method and the central difference method are used to discretize the space fractional Schrödinger equation and the space fractional Klein-Gordon equation in time direction, respectively. The Fourier spectral method is used to discretize the NCSFKGS equations in space direction. This numerical method conserves the mass and energy in the discrete level. The convergence of the numerical method is of second order accuracy in time and spectral accuracy in space. Rigorous proofs are developed here for the conservation laws and the convergence of the numerical method. Numerical experiments are presented to confirm the theoretical results and they proved that our numerical method is very efficient (it only takes a few seconds to get the numerical solutions). K E Y W O R D Sconvergence analysis, fast Fourier transform, Fourier spectral method, mass and energy conservation laws, nonlinear coupled space fractional Klein-Gordon-Schrödinger equations